TITLE: GEOMETRIC ATTITUDE CONTROL OF RIGID BODIES VIA CONTRACTION ON MANIFOLDS.
ABSTRACT: The attitude (or orientation) of an object is often crucial in its ability to perform a task,
whether the task is driving a car, flying an aircraft, orientating a satellite, or capturing the perfect
moment with a camera. While it is intuitive for humans to move (or control) the objects to the
proper attitude to perform these tasks, it is not trivial for autonomous systems. In traditional
control approaches, the attitude is often parameterized by Euler angles or quaternions which
exhibit problems such as gimbal lock or ambiguity in representation, respectively. This means
that controllers using these parameterizations cannot achieve global stability. More recent
works have achieved global stability by working directly on the configuration manifold, but are
generally complex and have many parameters to tune.
In this work, we present a simple geometric attitude controller with intuitive tuning parameters,
showing that it is locally exponentially stable and almost globally asymptotically stable; the
exponential convergence region is much larger than existing non-hybrid geometric controllers
(and covers almost the entire rotation space). The controller’s stability and convergence rate are
proved using contraction analysis combined with optimization. The key in this combination
is that the contraction metric is a linear matrix inequality with a special structure stemming
from the configuration manifold SO(3), and we show that the controller’s parameters can be
automatically tuned. On the way, we generalized results on Riemannian metrics for tangent
bundles to allow for a larger set of solutions.
In future work, we plan to improve the simple geometric attitude controller by introducing
a “feed-forward” term. The idea being that if the local controller converges exponentially to a
reference trajectory that is at the same time converging exponentially to a desired attitude, then
the whole system is globally exponentially stable. In addition, we plan to extend our framework
to use the stabilizing metric to implicitly define a controller that is point-wise (i.e., for a single
state and time) optimal; this extension is inspired by, but not a straightforward application of,
recent results on Control Lyapunov Functions and Control Barrier Functions.
COMMITTEE: ADVISOR Professor Roberto Tron, ME/SE; Professor Calin Belta, ME/SE/ECE; Professor John Baillieul, ME/SE/ECE; Professor Sean Andersson, ME/SE